Free-atom wave function

Although the calculated molecular parameters De = 3.15 eV, re = 1.64 a0 do not compare well with experiment the simplicity of the method is the more important consideration. Various workers have, for instance, succeeded to improve on the HL result by modifying the simple Is hydrogenic functions in various ways, and to approach the best results obtained by variational methods of the James and Coolidge type. It can therefore be concluded that the method has the correct symmetry to reproduce the experimental results if atomic wave functions of the correct form and symmetry are used. The most important consideration will be the effect of the environment on free-atom wave functions. 

Like electron spin, the valence state of an atom has no meaning in terms of free-atom wave functions. Like spin it could be added on by an ad hoc procedure, but this has never been achieved beyond the qualitative level. All conventional methods of quantitative quantum chemistry endeavour to simulate atomic behaviour in terms of free-atom functions. 

When the ionization spheres of two neighbouring atoms interpenetrate, their valence electrons become delocalized over a common volume, from where they interact equally with both atomic cores. The covalent interaction in the hydrogen molecule was modelled on the same assumption in the pioneering Heitler-London simulation, with the use of free-atom wave functions. By the use of valence-state functions this H-L procedure can be extended to model the covalent bond between any pair of atoms. The calculated values of interatomic distance and dissociation energy agree with experimentally measured values. 

Nevertheless this method has been successfully used by Van Vleck when establishing the nature of the effective Hamiltonian for a magnetic insulator [40]. In that case, the electrons involved in the magnetic properties are locahzed on each atom. In other words, an assumption of localization is made for each free-atom wave function its magnitude decreases exponentially when any electron is removed to a large distance from the center R m) of... 

An example, where this idea is directly exploited for the construction of the wave function is the atoms-in-molecules (AIM) method of Moffit [43, 44]. In this approach the ground and excited-state wave functions of the isolated atoms are used for the expansion of the total wave function. As noted by Schipper [45] this procedure is not flexible enough for a satisfactory description of the strong covalent bonds. The formation of the molecule cannot be considered as a small perturbation to the free-atom wave function, therefore highly excited and ionized atomic wave functions would also be needed for a satisfactory description of the molecule. 

Poltizer P, Parr RG, Murphy DR (1985) Approximate determination of Wigner-Seitz radii from free-atom wave functions. Phys Rev B 31(10) 6809-6812... 

Figure 1. The (augmented plane wave) wave function produced by a combination of atomic states with a free electron wave function between the ion cores."...

The concepts which we need for understanding the structural trends within covalently bonded solids are most easily introduced by first considering the much simpler system of diatomic molecules. They are well described within the molecular orbital (MO) framework that is based on the overlapping of atomic wave functions. This picture, therefore, makes direct contact with the properties of the individual free atoms which we discussed in the previous chapter, in particular the atomic energy levels and angular character of the valence orbitals. We will see that ubiquitous quantum mechanical concepts such as the covalent bond, overlap repulsion, hybrid orbitals, and the relative degree of covalency versus ionicity all arise naturally from solutions of the one-electron Schrodinger equation for diatomic molecules such as H2, N2, and LiH. 

The spin Hamiltonian can be obtained from the MO s in a manner similar to that used in Sec. III. In this case the parameters of the spin Hamiltonian are determined by the Cy/ s of the ground and excited state MO s as well as by the values of (E0 — E ), f, and av. In a complete calculation the values of cjt and (E0 — En) would be found by minimizing the total energy, but this is a difficult computation and has been attempted only infrequently. The most notable attempt in this direction is the calculation by Shulman and Sugano (24,25) on KNiF3. The general practice has been to determine values of (E0 — En) from optical spectra, from atomic spectra, and <>-3>av from free-ion wave functions and to use these values plus the experimental values of the spin Hamiltonian parameters to determine the values of the Cy/ s. 

The crystal field theory. The basics of the CFT were introduced in the classical work by Bethe [150] devoted to the description of splitting atomic terms in crystal environments of various symmetry. The splitting pattern itself is established by considering the reduction in the symmetry of atomic wave functions while the spatial symmetry of the system goes down from the spherical (in the case of a free atom) to that of a point group of the crystal environment. It is widely described in inorganic chemistry textbooks (seee.g. [152]). 

The quantum content of current theories of chemical cohesion is, in reality, close to nil. The conceptual model of covalent bonding still amounts to one or more pairs of electrons, situated between two atomic nuclei, with paired spins, and confined to the region in which hybrid orbitals of the two atoms overlap. The bond strength depends on the degree of overlap. This model is simply a paraphrase of the 19th century concept of atomic valencies, with the incorporation of the electron-pair conjectures of Lewis and Langmuir. Hybrid orbitals came to be introduced to substitute for spatially oriented elliptic orbits, but in fact, these one-electron orbits are spin-free. The orbitals are next interpreted as if they were atomic wave functions with non-radial nodes at the nuclear position. Both assumptions are misleading. 

It is surprising that a reasonable molecular wave function synthesized from two well-defined atomic wave functions should fail so comprehensively to account for the molecular properties. It shows that the atoms involved in formation of the molecule are not in their ground states. One way of improving the situation is in fact by using a linear combination of Is and 2pz functions to synthesize the molecular wave function. However, this procedure has no valid basis and cannot produce the final answer. The real reason for the failure is that atomic functions refer specifically to free atoms only. At... 

For free atoms this value can be calculated from the atomic wave-functions so that, to a first approximation, the p-electron density at the magnetic nucleus under study can be calculated from the ratio of the experimental and the atomic coupling constants. Furthermore, the direction of the largest component of the anisotropic coupling tensor coincides with the direction of the p-orbital. This is thus an important factor in the identification of the radical species. 

Fig. 37. Experimental values for Tb(OH)3 in the ordered state at 2.6K, after making extinction corrections. The broken curve is derived from nonrelativistic free ion wave functions. The solid curve is the best fit to the experimental points and falls directly on the relativistic free ion form factor. All curves are normalized to a magnetic moment of 8.9 jUs/Tb atom [after Ref. (757)]...

We consider two metallic free-electron systems, with atomically flat surfaces separated by vacuum over a distance Ax (Figure 20). In fact, the model system is an extension of the metal surface considered in Section 4.5. The complex potential energy barrier at a metal surface, discussed in Section 4.5 is simplified here to a rectangular barrier. We look for the quantum-mechanical probability that an electron in phase A is also present in phase B. This probability is given by the ratio of squared amplitudes, and A, of the free-electron wave function in phase B and A, respectively. It is quantified by the transmission coefficient ... 

For the case cited above, the ponderomotive energy is approximately 1 eV. For typical short pulse experiments today, this energy can easily be hundreds of electron volts. Therefore the wave function of a photoelectron in an intense laser field does not resemble that of the normal field-free Coulomb state, but is dressed by the field, becoming, in the absence of a binding potential, a Volkov state [5], This complex motion of the photoelectrons in the continuum is very difficult to reproduce in terms of the field-free atomic basis functions, so that we have chosen to define our electron wave functions on a finite difference grid. These numerical wave functions have the flexibility to represent both the bound and continuum states in the laser field accurately. 

As has been said before, plane waves might serve as general and straightforward basis functions if there were not the rapid oscillations of the atomic wave functions close to the nuclei. If these oscillations are artificially suppressed, such as in the free-electron model, plane waves are the optimum choice. Since... 

In a weak field, atomic considerations dictate which orbitals are filled, as was the case in the rare earths. Atomic states described by term symbols of total L and S describe the states for d orbital occupation these are spht by the d—d electrostatic repulsions. Spin orbital interaction is small in the transition metals and usually neglected in the first-order description of the levels. A weak crystal field shifts the levels and effects a spfitting which occurs because the crystal field removes the degeneracy of an L level the L level splits into its components. Atomic free ion wave functions having the S5unmetry of the crystal field are used to calculate the splittings. 

It is absolutely vital for our discussion to have a good definition of a localized electronic state as opposed to one which is delocalized. The later will also be referred to as a quasi-free electron. Our primary definition will be that the localized electron is characterized by a wave function which decays exponentially with distance the same type of definition used for simple atomic wave functions. The quasi-free electron... 

Metals. In simple metals such as the alkali and alkaline earth metals as well as A1 the valence electrons occupy only s and p levels. The rather extended shape of the atomic wave functions leads to a strong overlap and delocalization in the condensed phase. As a result, one obtains almost free-electron-like behavior for the electrons near the Fermi level and high electrical conductivity. Similarly, in the case of Cu and Ag, the electronic states at the Fermi level are derived from very delocalized s and p states, which explains the excellent electrical conductivity of these metals. Since the actual electrical conductivity of a material is strongly influenced by the scattering of the conduction electrons by impurities, lattice defects, and the thermal motion of the atomic nuclei, the quantitative prediction of electrical conductivity is difficult. 

Carlson, T. A. Lu, C. C. Tucker, T. C. Nestor, C. W. Malik, F. B. Eigenvalues, Radial Expectation Values, and Potentials for Free Atoms from Z = 2 to 126 as calculated from Relativistic Hartree-Fock-Slater Atomic Wave Functions, Oak Ridge National Laboratory, 1970, pp. 1-29. 

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